Consider two bonds:

Bond A maturing in October $2029,$ with a face value of $$1,000$ and a coupon rate of $5\mathrm{.}25\%$ and

Bond B maturing in October $2027,$ with a face value of $$1,250$ and a coupon rate of $6\mathrm{.}15\%.$

$($a$)$ Calculate duration and modified duration for the two bonds, assuming for Bond A a YTM of $5\%,$ and for Bond B a YTM of $6\%,$ and both bonds priced at par. How does the duration change for each bond, when YTM increases by $1$ percentage point? What is the percentage change in the price of bond due to $1$ percentage point change in yield for each bond?

$($b$)$ Now, suppose the quoted price of the bond is $$986$ for Bond A and $$1,105$ for Bond B$.$ How do your answers to $($a$)$ change?

$($c$)$ Assuming expected inflation in $2028$ and $2029$ of $2\mathrm{.}1\%$ and $1\mathrm{.}9\%,$ respectively, which bond is a better investment at quoted prices in $($b$)?$ Why?

Show your work $($preferred submission via Excel$).$